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A short proof of Eilenberg and Moore’s theorem

Maria Nogin (2007)

Open Mathematics

In this paper we give a short and simple proof the following theorem of S. Eilenberg and J.C. Moore: the only injective object in the category of groups is the trivial group.

Active sums I.

J. Alejandro Díaz-Barriga, Francisco González-Acuña, Francisco Marmolejo, Leopoldo Román (2004)

Revista Matemática Complutense

Given a generating family F of subgroups of a group G closed under conjugation and with partial order compatible with inclusion, a new group S can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group S is called the active sum of F, has G as a homomorph and is such that S/Z(S) ≅ G/Z(G) where Z denotes the center.The basic question we investigate in this paper is: when is the active sum S of the family F isomorphic to the...

On localizations of torsion abelian groups

José L. Rodríguez, Jérôme Scherer, Lutz Strüngmann (2004)

Fundamenta Mathematicae

As is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by | T | whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, we completely characterize the relationship...

On the existence of group localizations under large-cardinal axioms.

Carles Casacuberta, Dirk Scevenels (2001)


Uno de los problemas abiertos más antiguos de la teoría de grupos categórica es si todo par ortogonal (formado por una clase de grupos y una clase de homomorfismos que se determinan mutuamente por ortogonalidad en el sentido de Freyd-Kelly), se halla asociado a un funtor de localización. Se sabe que esto es cierto si se acepta la validez de un cierto axioma de cardinales grandes (el principio de Vopenka), pero no se conoce ninguna demostración mediante los axiomas ordinarios (ZFC) de la teoría de...

The Roquette category of finite p -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

Let p be a prime number. This paper introduces the Roquette category p of finite p -groups, which is an additive tensor category containing all finite p -groups among its objects. In p , every finite p -group P admits a canonical direct summand P , called the edge of P . Moreover P splits uniquely as a direct sum of edges of Roquette p -groups, and the tensor structure of p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from p to...

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